That move has fallen through, but Alexander's arrival at Greenwich is expected to be followed by more noteworthy signings in the next few weeks.

Alexander, who was born in Lambeth, started his career at West Ham United and was loaned out to Exeter City before the Hammers sold him to Swindon Town for £300,000 in July 2000.

He had a spell with Hull City before moving back to London, scoring 52 goals in 179 appearances for Orient between 2003 and 2007.

Alexander played 91 games for Millwall, netting 21 times, and then had two seasons at Brentford, scoring 21 goals in 62 appearances. He joined Crawley on loan in March 2012, later joining the club permanently and then serving as interim manager when Richie Barker was sacked last November.

He played seven times for Burton Albion at the end of last season before being released.

A Greenwich statement said: "It is a tremendous coup for a club playing at Step 5 to be able to attract such an experienced player from the Football League.

"The club are delighted to have signed a player with such a pedigree.

"Unfortunately, the attempt to sign Edgar Davids has not been as successful, but other signings are expected to be made in the near future."

Greenwich's pre-season schedule includes a week-long tour to Cyprus, which will see Ian Jenkins' side play friendlies against ENI Ipsonas and AEL Limassol in the Pareklishia Stadium. The tour is being sponsored by DGS Marine, the company owned and run by Borough president David Skinner.

Greenwich, who play their home games at Dartford's Princes Park, start their pre-season campaign with a friendly against Bridon Ropes at Coldharbour Leisure Centre on Friday.